Solving some problems of advanced analytical nature posed in the SIAM-review

In this paper, three SIAM-Review problems selected from Vol. 34 (1992) are reconsidered and treated using methods according to my own vision on them. 1 Consider the functions S(v) and C(v) defined as the sums of two infinite double series : S(v) = +∞ m=0 +∞ n=1 (−1) m+n sin(2v √ m 2 + n 2) √ m 2 + n 2 , (1.1) C(v) = +∞ m=0 +∞ n=1 (−1) m+n cos(2v √ m 2 + n 2) √ m 2 + n 2 , (1.2) whereby it is indifferent in which order of succession of m and n the summations are carried out on account of the symmetry of the summands with respect to m and n. Find closed expressions for S(v) and C(v) for arbitrary real v and try to deduce from them whether the conjectures S(v) = −v/2 if − π/ √ 2 < v < π/ √ 2 , C(v) = 0 if v = ±5/4 , (1.3) based upon numerical calculations, hold or not. These sums arose in finite-size scaling studies of the three-dimensional spherical model.